9 DETERMINACIONES ANALÍTICAS
9.4 Pruebas para la detección de agentes infecciosos transmisibles por transfusión
The validation of the proposed formulation is performed by comparing the obtained numerical results with the test and nonlinear finite element results for some basic but
typical discontinuity regions: simply supported deep beams with top and suspended loads, and continuous deep beams. Deep beams—top load
The presented methodology is compared with the well- known Leonhardt and Walther (1966) deep beam tests and with nonlinear finite element analysis using the ATENA program. Two simply supported beams (WT2 and WT3) in particular were analyzed. These results were previously presented by Lourenço et al. (2006) and later by Nunes (2008).
The geometrical dimensions, loads, and steel reinforce- ment layouts of Deep Beams WT2 and WT3 are represented in Fig. 7. Tests have shown considerable redistribution of internal stresses due to cracking. Experimental ultimate loads reached Pu = 1.28pu = 1195 kN (268.6 kips) and Pu = 1290 kN
(290.0 kips) for Deep Beams WT2 and WT3, respectively. At rupture, Deep Beam WT2 exhibited yielding of the bottom reinforcement and maximum support pressures of approxi- mately 1.06fcu. With regard to Deep Beam WT3, bottom steel
stresses of approximately 370 MPa (53.7 ksi) and support pressures of approximately 1.19fcu were measured (fcu—
compression strength of 200 x 200 x 200 mm3 [7.87 x 7.87 x
7.87 in.3] cubes; f
1c= 0.89fcu according to ACI 318-08). Steel
reinforcing detailing at the node region, where U-loops were adopted, improved concrete confinement and thus a higher local concrete strength was achieved.
Each developed stress field model (Fig. 8(a) and (c)) simulates all horizontal deep beam reinforcements, and the adaptive variables are node horizontal coordinates of the web reinforcement.
The numerical results agree quite well with tests at all load stages (Fig. 8(b) and (d)). As mentioned previously, the reinforcing detailing at the node region, where U-loops were adopted, improved concrete confinement and increased concrete strength at the supports. This aspect was disre- garded in the numerical model, leading to ultimate load values slightly lower than the experimental results (Pu =
1100 kN [247.3 kips]). Concerning the slight deviations of the mean steel strains between the numerical analysis and experimental results, it must be kept in mind that the steel strains were measured by strain gauges glued on bars, although the test outputs depend on the crack location and strain gauge positions.
It should be pointed out that the stiffness reduction related to cracking of the main tie induces significant internal stress redistributions, evident in the stress field model configu- rations (Fig. 8(a) and (c)) or in the internal lever arm (z/l)
Fig. 6—Adaptive analysis of test example: (a) force versus displacement; (b) force versus stresses at Element 1; and (c) force versus model configuration. (Note: 100 MN = 22,481 kips; 0.1 m = 3.94 in.; 100 MPa = 14.5 ksi.)
Fig. 7—Deep beam geometry and reinforcement layout (Leonhardt and Walther 1966): (a) WT2; and (b) WT3.
Fig. 8—(a) Deep Beam WT2 adaptive stress field model for several load steps; (b) test, nonlinear finite element, and adaptive stress field results for Deep Beam WT2; (c) Deep Beam WT3 adaptive stress field model for several load steps; and (d) test, nonlinear finite element, and adaptive stress field results for Deep Beam WT3. (Note: 100 kN = 22.5 kips; 1 mm = 0.039 in.)
Fig. 9—Deep Beams WT2 and WT3 numerical and test crack pattern for ultimate load: (a) WT2: test (Leonhardt and Walther 1966); (b) WT2: adaptive stress field model; (c) WT2: finite element analysis; (d) WT3: test (Leonhardt and Walther 1966); (e) WT3: adaptive stress field model; and (f) WT3: finite element analysis.
variation. These aspects are particularly relevant for the Deep Beam WT2 test, where the available ductility allowed full use of the beam effective depth. Figure 9 shows the agreement of the numerical and test crack patterns for ultimate load.
In general, the numerical results agreed quite well with those obtained experimentally. The force-strain and force- crack width curves obtained by the adaptive stress field analysis and the finite element method closely follow the test at different stages—namely, uncracked, cracked, and post- yielding. In this study, the finite element method provides slightly stiffer results, evident in the force-displacement curves; however, higher crack widths were obtained. The Adaptive Stress Field Model approach was revealed to be quite appropriate and, furthermore, the graphical output allowed an excellent visualization of structural behavior for all load stages.
Deep beams—suspended load
Two simply supported deep beams with suspended loads (WT6 and WT7) were also tested by Leonhardt and Walther (1966) and numerically compared with the obtained results. Deep beam geometrical dimensions, loads, and steel reinforcement layouts are represented in Fig. 10(a) and (b). Identical dimensions and reinforcement layouts of Deep Beams WT2 and WT3 were adopted. The web vertical reinforcement suspends the applied loads at the bottom surface of the deep beams.
Three layers of reinforcement were considered in the stress field models. The top distributed reinforcement layers were discarded because little influence on the global analysis was observed. The adaptive variables are the node horizontal position of the bottom web reinforcement and the node vertical position of one of the vertical reinforcements.
Such as the deep beams with top loads, the stiffness reduc- tion related to cracking of the bottom tie leads to internal
stress redistribution, clearly shown in the stress field model configurations illustrated in Fig. 10(c) and (d). Again, the available ductility allowed the deep beam with less bottom reinforcement to fully exploit the effective depth. The numer- ical mean strain results showed good agreement with the experimental output, especially in the uncracked stage and after cracking of the deep beam bottom surface (Fig. 11(a) and (b)). However, a significant discrepancy was obtained for the numerical and test crack widths because the observed test crack widths were measured in the deep beam front surface not effectively controlled by the main reinforcement.
Figure 11(c) to (f) shows the agreement of the numerical stress field distribution and test crack pattern for ultimate load. Continuous deep beam
A two-span continuous Deep Beam DWT2 (Leonhardt and Walther 1966) was also analyzed, and the numerical and test results were compared. Figure 12(a) presents the beam test geometry and loads. The main and distributed reinforce- ments of the deep beam were simulated in the adaptive stress field analysis. The adaptive variables were the horizontal location of the nodes leading to tension or compression in horizontal elements.
Figure 12(b) illustrates the comparison between the numerical and experimental results. A reasonable agreement should, in general, be noticed. The model thickness enlarge- ment of the central support region all along the beam height was not simulated in the numerical model, which seems to be more flexible; however, the initial elastic range of the numerical model closely follows the linear results obtained with the finite element analysis.
Figure 12(c) and (d) shows the good agreement between the test crack pattern and the numerically obtained stress field distribution. Because the thickness enlargement of the
Fig. 10—Deep beam geometry and reinforcement layout (Leonhardt and Walther 1966): (a) WT6; and (b) WT7. Adaptive stress field model for several load steps: (c) WT6; and (d)
WT7. (Note: 1 m = 39.4 in.; 100 mm2 = 0.155 in.2; 100 kN = 22.5 kips.)
Fig. 11—Comparison of numerical and test results: (a) WT6; and (b) WT7. Stress field configuration and test ultimate load crack pattern (Leonhardt and Walther 1966): (c) and d) WT6; and (e) and (f) WT7. (Note: 100 kN = 22.5 kips.)
Fig. 12—(a) Deep Beam DWT2 geometry and reinforcement layout (Leonhardt and Walther 1966); (b) Deep Beam DWT2 tests and adaptive stress field results; (c) Deep Beam DWT2 test crack pattern (Leonhardt and Walther 1966); and (d) adaptive stress field model.
(Note: 1 m = 39.4 in.; 100 mm2 = 0.155 in.2; 100 kN = 22.5 kips; 100 MPa = 14.5 ksi.)
middle support was not simulated, the numerically obtained stress field inclination is slightly flatter.
CONCLUSIONS
This study aimed to provide a step forward in the applica- tion of the stress field models for the analysis and design of structural concrete. The proposed technique, Adaptive Stress Field Models, employed the convenient simplifications inherent to the stress field models to develop a tailor-made tool for structural concrete. The method was implemented in a computer program; however, the idea was not to build up a fully automatic tool, even less a “black box.” One of the main aspects is the visualization of the flow of forces provided by the method, which offered a unique awareness of the struc- tural behavior along the loading process—essential for an adequate judgment of the outputs. The main features of the proposed approach are as follows:
• The internal stress redistributions due to the nonlinear behavior of the materials were accomplished by the incorporation of the adaptive structures concept to stress field systems—in particular, model follows energy. • The model configuration at each load step was thus
obtained following the least complementary energy. It intends to consistently follow the stress field model concept by setting an initial stress field distribution and selecting the appropriate variables that will be adjusted: geometry and/or forces of the model. The need to select a model to start the tool should not be considered a drawback; rather, it should be considered a support for the engineer to understand the structure’s main behavior before starting a more complex analysis.
• The mechanical properties of the compression and tension elements were directly obtained from the geom- etry of the stress fields, accounting for the nonlinear constitutive relationships of the materials.
• Special attention was given to the behavior of reinforced ties: the well-known constitutive relationship proposed in technical documents, mainly defined for stabilized cracking, can be inappropriate for some discontinuity regions. The cracking pattern is sometimes character- ized by a main crack that deeply influences global struc- tural behavior. It was assumed that this phenomenon was related to the tie stress distribution, preventing the formation of a stabilized cracking.
• The decrease in the concrete compressive strength due to its transverse strain state is also considered in the formu- lation to predict an eventual premature concrete crushing. The technique was validated by comparing the numerical results with the outcomes from the tests and nonlinear finite element analysis. The following aspects should be pointed out: • The internal stress redistributions after cracking
observed in several of the presented tests were well- simulated in the numerical model. In fact, the stiffness reduction related to cracking of the main ties induced significant stress redistributions—evident in the stress field model configurations.
• The force-strain and force-crack width curves obtained by the adaptive stress field analysis closely followed the test results at different stages (cracked and post-yielding). • The model approach was revealed to be quite appro- priate and, furthermore, the graphical output allowed an excellent visualization of structural behavior for all load stages.
• Finally, in all cases, it was considered that the main struc- tural behavior aspects were well-simulated, showing the capability of the methodology for the nonlinear analysis of structural concrete regions.
NOTATION
(matrixes and vectors in bold; subscript notation follows primary notation) A = area
AT = transpose of matrix A that represents node incidences of elements
a, b = widths
Dc, Ds = matrixes with cosines and sines, respectively, of each bar angle
E = elastic modulus
F = applied forces
F = vector of applied forces
f = material strength
l = length
M = moment
N = axial force
N = vector of axial forces
P = applied forces
q = distributed load
r = radius
T = condensation of matrixes Tc and Ts
Tc, Ts = condensation of matrixes ATDc and ATDs, respectively
t = thickness u, U = complementary energy V = domain v = variable w = crack width x, y = coordinates
z = inner level arm a, b, h = coefficients D = elongation d = displacement e = strain q = angle r = reinforcement ratio (r = As/Ac) s = stress t = tangential stress Ø = diameter subscripts
1c = uniaxial compression strength of concrete
b = bond c = concrete, compression d = design i, j = constant k = characteristic m = mean, medium r = radius, radial s = steel t = tension, tangential u = ultimate w = web, wedge y = yielding REFERENCES
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